Boettcher Coordinate
Robert P. Munafo, 2023 Jul 18.
The Böttcher coordinate is the value related to the external angle (or external argument) of a point emerging from the iteration orbit dynamics of some iteration formula. It is defined along with a "Böttcher's equation" in the 1993 Carleson and Gamelin book Complex Dynamics. With regards to the Mandelbrot set specifically, it is cited in papers by Buff and Romera, as well as this wikibooks article).
The wikibooks article uses a different product "series" from the Romera paper. Romera 2013 states:
The computer programs to draw external rays of the Mandelbrot set use the Böttcher coordinate given by 4
Φ(c) = c PRODUCT(n=1..infinity) [1 + c/(fcn-1(c))2]1/2n (1a)
where c is the complex coordinate of a point outside ℳ and fc0(c)=c, fc1(c)=c2+c, fc2(c)=(c2+c)2+c, ... are the iterates of fc(z)=z2+c from the initial value z0=0.
The wikibooks article, citing Wolf Jung without a specific reference, states:
To compute Boettcher coordinate w use this formula:
w = Φc(z) = z PRODUCT(n=1..infinity) [fcn(z)/(fcn(z)-c)]1/2n (1b)
It looks "simple", but: square root of complex number gives two values so one have to choose one value; and more precision is needed near Julia set coordinate
In this second formula (1b), the terms of the infinite product have fcn(z)-c in the denominator, where fcn(z) simply refers to the iterates zn+1. Therefore the first z is z1 by the iterates definition, and fcn(z)-c is z(n+1)-c.
Since z(n+1) is zn2+c, we can change z(n+1)-c to zn2. Formula (1b) becomes:
z1 PRODUCT(n=1..infinity) [zn+1/zn2]1/2n
= z1 (z2/z12)(1/2) (z3/z22)(1/4)
(z4/z32)(1/8) ...
Since z1=c, z2=c2+c, z3=(c2+c)2+c, and so on, it becomes:
c ((c2+c)/c2)1/2 (((c2+c)2+c)/(c2+c)2)1/4 ((((c2+c)2+c)2+c)/((c2+c)2+c)2)1/8 ... (2)
Now in each term we have something like (bigpoly2+c)/(bigpoly2) so we can turn this into the sum of two fractions (bigpoly2)/(bigpoly2) + c/(bigpoly2), the first of which is 1 so the whole term is just 1+c/(bigpoly2):
c (1 + c/c2)1/2 (1 + c/(c2+c)2)1/4 (1 + c/((c2+c)2+c)2)1/8 ... (2)
Recalling again that z1=c, z2=c2+c, etc. we can put it back into product form as
Φ(c) = c PRODUCT(n=1..infinity)[(1+c/(zn2))2n]
with zn defined as in the iterates article. Now compare this to (1a) above, and remember that the indices zn are off by 1 as compared to the fcn-1(c)
Working backwards to (2) then continuing, we can work towards a series of nested square roots:
Φ(c) = c sqrt( (1+c/c2)
sqrt( (1+c/(c2+c)2)
sqrt( (1+c/((c2+c)2+c)2)
... ) ) )
= c ((c2+c)/c2)1/2
(((c2+c)2+c)/(c2+c)2)1/4
((((c2+c)2+c)2+c)/((c2+c)2+c)2)1/8
...
= c (z2/(z2-c)1/2
(z3/(z3-c)1/4
(z4/(z4-c)1/8
...
= c sqrt(z2/(z2-c) sqrt(z3/(z3-c)
sqrt(z4/(z4-c) ... ) ) )
We can see from these expansions that if c has a period then the Böttcher coordinate Φ(c) is undefined because of a division by zero, for example if z4=c then (z4-c) is zero.
Φ(c) is a complex number; as such it has a Magnitude Φ(c), and an Angle arg(Φ(c)). The "external ray" for a given angle a is the set of all points c for which arg(Φ(c))/π=a.
revisions: 20230718 first version (content taken from External Angle article)
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.
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